Abelian variety isogeny class 1.397.af over $\F_{397}$

• Label: 1.397.af
• Base field: $\F_{397}$
• Dimension: $1$
• $p$-rank: $1$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{397}$
Dimension:	$1$
L-polynomial:	$1 - 5 x + 397 x^{2}$
Frobenius angles:	$\pm0.459955669257$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-1563}) \)
Galois group:	$C_2$
Jacobians:	$6$

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$1$
Slopes:	$[0, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$393$	$158379$	$62576604$	$24840320739$	$9861713266533$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$393$	$158379$	$62576604$	$24840320739$	$9861713266533$	$3915101724970176$	$1554295350548326569$	$617055253377773165475$	$244970935600629522947148$	$97253461433811783746649939$

Jacobians and polarizations

This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{397}$.

Endomorphism algebra over $\F_{397}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-1563}) \).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
1.397.f	$2$	(not in LMFDB)